Quantitative Unique Continuation on Simplex and $\mathbb{Z}^n$
In this work, we establish the quantitative unique continuation on both $n$-dimensional simplex lattice $\Delta^{(n)}_{N}$ and $\mathbb{Z}^n$ for arbitrary $n\geq 3$.
math.AP↗
arXiv subjects
Shihe Liu
Publications and source records attributed to Shihe Liu.
Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$
In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.
math.SP↗
Anderson Localization for the hierarchical Anderson-Bernoulli model on $\mathbb{Z}^d$
In this paper, we prove Anderson localization for a hierarchical Anderson-Bernoulli model on lattice with arbitrary dimension, where the potential is characterized by a geometric hierarchical structure combined with fluctuations induced by independent and identically distributed (i.i.d.) Bernoulli random variables. Our method is also applicable to proving a probabilistic unique continuation result on $\mathbb{Z}^d$.
math.AP↗
On localization for the alloy-type Anderson-Bernoulli model with long-range hopping
In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on $\mathbb{Z}^d$ with exponential long-range hopping. This extends the work of Bourgain [Geometric Aspects of Functional Analysis, LNM 1850: 77--99, 2004], in which he pioneered a novel multi-scale analysis to treat Bernoulli random variables. Our proof is mainly based on Bourgain's method. However, to establish the initial scales Green's function estimates, we adapt the approach of Klopp [Comm. Math. Phys, Vol. 232, 125--155, 2002], which is based on the Floquet-Bloch theory and a certain quantitative uncertainty principle. Our proof also applies to an analogues model on $\mathbb{R}^d.$
math-ph↗
Extended states for the Random Schr\"odinger operator on $\mathbb{Z}^d$ ($d\geq 5$) with decaying Bernoulli potential
In this paper, we investigate the delocalization property of the discrete Schr\"odinger operator $H_\omega=-\Delta+v_n\omega_n\delta_{n,n'}$, where $v_n=\kappa |n|^{-\alpha}$ and $\omega=\{\omega_n\}_{n\in\mathbb{Z}^d}\in \{\pm 1\}^{\mathbb{Z}^d}$ is a sequence of i.i.d. Bernoulli random variables. Under the assumptions of $d\geq 5$, $\alpha>\frac14$ and $0<\kappa\ll1$, we construct the extended states for a deterministic renormalization of $H_\omega$ for most $\omega$. This extends the work of Bourgain [{\it Geometric Aspects of Functional Analysis}, LNM 1807: 70--98, 2003], where the case $\alpha>\frac13$ was handled. Our proof is based on Green's function estimates via a $6$th-order renormalization scheme. Among the main new ingredients are the proof of a generalized Khintchine inequality via Bonami's lemma, and the application of the fractional Gagliardo-Nirenberg inequality to control a new type of non-random operators arising from the $6$th-order renormalization.
math-ph↗
Anderson localized states for the nonlinear Maryland model on $\mathbb{Z}^d$
In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model $H=\varepsilon\Delta+\cot\pi(\theta+j\cdot\alpha)\delta_{j,j'}$ on $\mathbb{Z}^d$. Specifically, if $\varepsilon,\delta$ are sufficiently small, we construct a large number of time quasi-periodic and space exponentially decaying solutions (i.e., Anderson localized states) for the equation $i\frac{\partial u}{\partial t}=Hu+\delta|u|^{2p}u$ with a Diophantine $\alpha$. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.
math.AP↗